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체스를 20년 동안 놀고 있어서 놀기를 아주 잘 해요.

In this sentence, '놀고' is not appropriate expression for 'playing chess'We usually use '두다' for especially playing chess like games instead of '놀다', but we can also use just '하다 (to do in English)체스를 두다 체스를 게임을 하다체스를 (게임을) 하다.

So you got the correction:난 체스를 20년 동안 해봐서 아주 잘 해요.

You understand this quite well.

It uses the -아/어/여 보다 to describe "playing chess for 20 years". I think of that as to mean "to try doing something" but also to mean "to have the experience of doing something". - correct.

This sentence'저는 4달 동안 한국어를 공부하고 있어요.' is correct.

but"저는 4달 동안 한국어를 공부해 봐요."sounds odd because ~봐요 means 'let us do something' for suggestion or giving an instruction.If you can say 우리 4달 동안 한국어를 공부해 봐요 it sounds perfect but it means 'Let's study Korean for 4 months'. or, 'I will teach you Korean for 4 months.'

So, the problem occurred with 'vocabulary' not with 'grammar'Your grammar level seems very high, I am sure.